Degree Spectra of Differentially Closed Fields Russell Miller - PowerPoint PPT Presentation

Degree Spectra of Differentially Closed Fields Russell Miller Queens College & CUNY Graduate Center Recursion Theory Seminar University of California – Berkeley 14 April 2014 Joint work with Dave Marker. Russell Miller (CUNY) Spectra of DCF 0 Recursion Theory Seminar 1 / 15

Spectra of Countable Structures Let S be a structure with domain ω , in a finite language. Definition The Turing degree of S is the join of the Turing degrees of the functions and relations on S . If these are all computable, then S is a computable structure . Definition The spectrum of S is the set of all Turing degrees of copies of S : Spec ( S ) = { deg ( M ) : M ∼ = S & dom ( M ) = ω } . So the spectrum measures the level of complexity intrinsic to the structure S . Russell Miller (CUNY) Spectra of DCF 0 Recursion Theory Seminar 2 / 15

Facts About Spectra Theorem (Knight 1986) For all countable structures S but the automorphically trivial ones, the spectrum of S is upwards-closed under Turing reducibility. Many interesting spectra can be built using graphs, including upper cones, α -th jump cones { d : d ( α ) ≥ T c } , and more exotic sets of Turing degrees. (Greenberg, Montalb´ an, and Slaman recently constructed a graph whose spectrum contains exactly the nonhyperarithmetic degrees.) Indeed, graphs are complete , in the following sense: Theorem (Hirschfeldt-Khoussainov-Shore-Slinko 2002) For every countable structure S in any finite language, there exists a countable graph G which has the same spectrum as S . Russell Miller (CUNY) Spectra of DCF 0 Recursion Theory Seminar 3 / 15

Spectra of Algebraically Closed Fields Russell Miller (CUNY) Spectra of DCF 0 Recursion Theory Seminar 4 / 15

Spectra of Algebraically Closed Fields { all Turing degrees } . Russell Miller (CUNY) Spectra of DCF 0 Recursion Theory Seminar 4 / 15

Differentially Closed Fields A differential field is a field along with a differential operator δ on the field elements, respecting addition ( δ ( x + y ) = δ x + δ y ) and satisfying the product rule δ ( x · y ) = ( x · δ y ) + ( y · δ x ) . Such a field K is differentially closed if it also satisfies the Blum axioms : for all differential polynomials p , q ∈ K { Y } , ord ( q ) < ord ( p ) = ⇒ ( ∃ x ∈ K ) [ p ( x ) = 0 & q ( x ) � = 0 ] , where the order r = ord ( p ) is the largest derivative δ r Y used in p . This theory DCF 0 is complete and decidable and has quantifier elimination. Moreover, it has computable models: Theorem (Harrington, 1974) For every computable differential field k , there exists a computable model K of DCF 0 and a computable embedding g of k into K such that K is a differential closure of the image g ( k ) . Russell Miller (CUNY) Spectra of DCF 0 Recursion Theory Seminar 5 / 15

Noncomputable Differentially Closed Fields By analogy to ACF 0 , one may guess that all countable models of DCF 0 have computable presentations. However, it is known that there exist 2 ω -many (non-isomorphic) countable models of DCF 0 . Indeed: Theorem (Marker-M.) = DCF 0 with For every countable graph G , there exists a countable K | Spec ( K ) = { d : d ′ can enumerate the edges in some G ∗ ∼ = G } . Russell Miller (CUNY) Spectra of DCF 0 Recursion Theory Seminar 6 / 15

Noncomputable Differentially Closed Fields By analogy to ACF 0 , one may guess that all countable models of DCF 0 have computable presentations. However, it is known that there exist 2 ω -many (non-isomorphic) countable models of DCF 0 . Indeed: Theorem (Marker-M.) = DCF 0 with For every countable graph G , there exists a countable K | Spec ( K ) = { d : d ′ can enumerate the edges in some G ∗ ∼ = G } . It is not difficult to show that, for every G , there is another graph H s.t. { d : d ′ enumerates the edges in some G ∗ ∼ = G } = { d : d ′ ∈ Spec ( H ) } , and that conversely, for each H , there is some such G . So the theorem proves that every countable graph H yields a K | = DCF 0 with Spec ( K ) = { d : d ′ ∈ Spec ( H ) } . Russell Miller (CUNY) Spectra of DCF 0 Recursion Theory Seminar 6 / 15

Coding a Graph G into K | = DCF 0 Start with a copy ˆ Q of the differential closure of Q . Let A be the following infinite set of indiscernibles in ˆ Q : Q : δ y = y 3 − y 2 & y � = 0 & y � = 1 } . A = { a 0 , a 1 , . . . } = { y ∈ ˆ Each a m ∈ A will represent the node m from G . Let E a m a n be the elliptic curve defined by the equation y 2 = x ( x − 1 )( x − a m − a n ) . Q ) 2 are algebraic over The coordinates of all solutions to this curve in (ˆ Q � a m + a n � and E a m a n forms an abelian group, with exactly j 2 j -torsion points for every j , and with no non-torsion points. There is a homomorphism of differential algebraic groups from E a m a n into a vector group, whose kernel E ♯ a m a n is called the Manin kernel of E a m a n . Russell Miller (CUNY) Spectra of DCF 0 Recursion Theory Seminar 7 / 15

Coding a Graph G into K | = DCF 0 For each m < n with an edge in G from m to n , add a generic point of E ♯ a m + a n to our differential field. The coordinates of this point will each be transcendental over Q � a m + a n � . Let K be the differential closure of the resulting differential field. Thus the coding is: G has an edge from m to n ⇐ ⇒ ( ∃ ( x , y ) ∈ E ♯ a m a n )[ x is transcendental over Q � a m + a n � ] . In particular, the points we added do not accidentally give rise to any transcendental solutions to any other E ♯ a m ′ a n ′ . Russell Miller (CUNY) Spectra of DCF 0 Recursion Theory Seminar 8 / 15

Spec ( K ) = { d : d ′ enumerates some G ∗ ∼ = G } Now if d is the degree of a copy K ∗ ∼ = K , then with a d ′ -oracle, we enumerate the edges in some G ∗ as follows. Find all elements a ∗ m of the set A ∗ of indiscernibles in K ∗ , go through all solutions to E a ∗ n for m a ∗ each m < n , and ask whether each is transcendental over Q � a ∗ m , a ∗ n � and lies in E ♯ n . If we ever get an answer ”YES,” we enumerate ( m , n ) a ∗ m a ∗ into the edge relation of the graph G ∗ . Thus G ∗ ∼ = G : the isomorphism comes from restricting the isomorphism K ∗ → K to A ∗ → A . Conversely, if D ∈ d and D ′ enumerates the edges in some G ∗ ∼ = G , we build K ∗ ∼ = K using a d -oracle. Start building ˆ Q ∗ , finitely much at each step. At stage s , if it appears (from D ) that D ′ has enumerated an edge ( m , n ) in G ∗ , add a point x mn ∈ E ♯ n which is not (yet) algebraic a ∗ m a ∗ over Q � a m , a n � . If D ′ later changes and wipes out this enumeration, we can still make x mn a t -torsion point for some large t , hence algebraic. Finally, use Harrington’s theorem to build a D -computable differential closure K ∗ of the D -computable differential field defined here. Russell Miller (CUNY) Spectra of DCF 0 Recursion Theory Seminar 9 / 15

Low and Nonlow Degrees For every d ′ > 0 ′ , there exists a graph G such that d ′ enumerates a copy of G , but 0 ′ does not. Therefore: Corollary For every nonlow degree d (i.e., with d ′ > 0 ′ ), there exists some K | = DCF 0 of degree d such that K is not computably presentable. We now prove the converse: Theorem (Marker-M.) Every low model of DCF 0 is isomorphic to a computable one. This recalls the famous theorem of Downey-Jockusch that every low Boolean algebra is isomorphic to a computable one. Russell Miller (CUNY) Spectra of DCF 0 Recursion Theory Seminar 10 / 15

Principal Types over k Over a field E , the principal 1-types are generated by the formulas p ( X ) = 0, where p ∈ E [ X ] is irreducible. Over a differential field k , this is not enough! Over Q , the differential polynomial ( δ Y − Y ) is irreducible, but only the following formula generates a principal type: δ Y − Y = 0 & Y � = 0 . In general, we need pairs ( p , q ) from k { Y } , with ord ( p ) > ord ( q ) . If the formula p ( Y ) = 0 � = q ( Y ) generates a principal type, then ( p , q ) is a constrained pair , and p is constrainable . Every principal type is generated by a constrained pair, but not all irreducible p ( Y ) are constrainable. p ( Y ) = δ Y is a simple counterexample. Fact p ∈ k { Y } is constrainable ⇐ ⇒ p is the minimal differential polynomial of some x in the differential closure K of k . It is Π k 1 for ( p , q ) to be constrained, and Σ k 2 for p to be constrainable. Russell Miller (CUNY) Spectra of DCF 0 Recursion Theory Seminar 11 / 15

Low Differentially Closed Fields K If K is low, then the (computable infinitary) Π 0 1 -theory of K has degree 0 ′ , hence is computably approximable. This allows us to “guess” effectively at the minimal differential polynomial of any x ∈ K over the differential subfield Q � x i 0 , . . . , x i n � ⊆ K generated by an arbitrary finite tuple from K . Writing K = { x 0 , x 1 , . . . } and guessing thus, we build a computable differential field F = { y 0 , y 1 , . . . } and finite partial maps h s : K → F such that: ( ∀ n ) lim s h s ( x n ) exists; and ( ∀ m ) lim s h − 1 s ( y m ) exists; and ∀ s h s is a partial isomorphism, based on the approximations in K to the minimal differential polynomials of its domain elements. Thus h = lim s h s will be an isomorphism from K onto F . Russell Miller (CUNY) Spectra of DCF 0 Recursion Theory Seminar 12 / 15